A129533 Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 12, 6, 0, 0, 10, 30, 30, 10, 0, 0, 15, 60, 90, 60, 15, 0, 0, 21, 105, 210, 210, 105, 21, 0, 0, 28, 168, 420, 560, 420, 168, 28, 0, 0, 36, 252, 756, 1260, 1260, 756, 252, 36, 0, 0, 45, 360, 1260, 2520, 3150, 2520, 1260, 360, 45, 0, 0, 55, 495

Offset

0,8

Comments

Previous name was: Triangle read by rows: T(n,k)=derivative of the q-binomial coefficient [n,k] evaluated at q=1 (0<=k<=n). - N. J. A. Sloane, Jan 06 2016

For example, T(5,2)=30 because [5,2] = q^6 + q^5 + 2*q^4 + 2*q^3 + 2*q^2 + q + 1 with derivative 6q^5 + 5q^4 + 8q^3 + 6q^2 + 4q + 1, having value 30 at q=1. - Emeric Deutsch, Apr 22 2007

Sum of entries in n-th antidiagonal = n(n-1)2^(n-3) = A001788(n-1).

T(n,k) = A094305(n-2, k-1) for n >= 2, k >= 1.

T(n,k) is total number of pips on a set of generalized linear dominoes with n cells (rather than two) and with the number of pips in each cell running from 0 to k (rather than 6). T(2,6) = 168 gives the total number of pips on a standard set of dominoes. We regard a generalized linear domino with n cells and up to k pips per cell as an ordered n-tuple [i_1, i_2, ..., i_n] with 0 <= i_1 <= i_2 <= ... <= i_n <= k. - Alan Shore and N. J. A. Sloane, Jan 06 2016

T(n,k) can also be written more symmetrically as the trinomial coefficient (n+k; n-1, k-1, 2). - N. J. A. Sloane, Jan 06 2016

As a triangle read by rows, T(n,k) is the total number of inversions over all length n binary words having exactly k 1's. T(n,k) is also the total area above all North East lattice paths from the origin to the point (k,n-k). - Geoffrey Critzer, Mar 22 2018

References

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

Links

Table of n, a(n) for n=0..68.

Index entries for sequences related to dominoes

Formula

T(n,k) = (1/2)*k*(k+1)*binomial(n,k+1).

G.f.: G(q,z) = qz^2/(1-z-qz)^3.

Example

Array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... (A000004)
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, ... (A000217)
0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, ... (A027480)
0, 6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, ... (A033487)
0, 10, 60, 210, 560, 1260, 2520, 4620, 7920, 12870, ... (A266732)
0, 15, 105, 420, 1260, 3150, 6930, 13860, 25740, ... (A240440)
0, 21, 168, 756, 2520, 6930, 16632, 36036, ... (A266733)
...
If regarded as a triangle, this begins:
  0;
  0,  0;
  0,  1,  0;
  0,  3,  3,  0;
  0,  6, 12,  6,  0;
  0, 10, 30, 30, 10,  0;
  0, 15, 60, 90, 60, 15, 0;
  ...

Maple

dd:=proc(n, m) if m=0 or n=0 then 0 else (m+n)!/(2*(m-1)!*(n-1)!); fi; end;
f:=n->[seq(dd(n, m), m=0..30)];
for n from 0 to 10 do lprint(f(n)); od: # produces sequence as square array
T:=(n, k)->k*(k+1)*binomial(n, k+1)/2: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

Mathematica

Table[Table[D[Expand[FunctionExpand[QBinomial[n, k, q]]], q] /. q -> 1, {k, 0, n}], {n, 0, 15}] // Grid (* Geoffrey Critzer, Mar 22 2018 *)

Crossrefs

Cf. A001788.

Rows give A000004, A000217, A027480, A033487, A266732, A240440, A266733.

A128503 and A094305 are very similar sequences.

Sequence in context: A299602 A299554 A300175 * A360849 A155999 A338034

Adjacent sequences: A129530 A129531 A129532 * A129534 A129535 A129536

Keyword

nonn,tabl

Author

Emeric Deutsch, Apr 22 2007

Extensions

Entry revised by N. J. A. Sloane, Jan 06 2016

Status

approved